** Speaker:** Angelica Cueto, Ohio State University

** Title:** Tropical geometry of genus 2 curves

** Abstract: **
In this talk, I will discuss the structure of tropical and non-Archimedean analytic genus 2 curves and their
moduli from three perspectives: as 2-to-1 covers of $P^1$ branched at 6 points, as solutions to the hyperelliptic
equation $y^2=f(x)$ where $f$ has degree 5 and as metric graphs dual to genus 2 nodal algebraic curves over a
valued field. Our first description allows us to give an explicit combinatorial rule to characterize each such
metric graph together with a harmonic map to a metric tree on 6 leaves, in terms of the valuations of 6 branch
points in $P^1.$ Even though the tropicalization of a plane hyperelliptic curve shows no genus, we provide
explicit re-embeddings of the input planar hyperelliptic curve that reveals the correct metric graphs.

From our third viewpoint, we consider the moduli space of abstract genus two tropical curves and translate the classical Igusa invariants characterizing isomorphism classes of genus two algebraic curves into the tropical realm. While these tropical Igusa functions do not yield coordinates on the tropical moduli space, we propose an alternative set of invariants that provides new length data. This is joint work with Hannah Markwig.